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! Degrees
! Degrees
! Cents
! Cents
! Approximate Ratios *
! Approximate Ratios<ref group="note">{{sg|limit=17-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO interval names and notation]]
! colspan="3" |[[SKULO interval names|SKULO interval names and notation]]
Line 47: Line 47:
| perfect unison
| perfect unison
| D
| D
|D
| D
|-
|-
| 1
| 1
Line 58: Line 58:
| comma-wide unison, large unison
| comma-wide unison, large unison
| KD, LD
| KD, LD
|KD
| KD
|-
|-
| 2
| 2
Line 69: Line 69:
| super unison, on unison
| super unison, on unison
| SD, OD
| SD, OD
|SD
| SD
|-
|-
| 3
| 3
Line 80: Line 80:
| uber unison, hyper unison, hypominor 2nd
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, HD, uEb
|UD, uEb
| UD, uEb
|-
|-
| 4
| 4
Line 91: Line 91:
| classic aug unison, subminor 2nd
| classic aug unison, subminor 2nd
| kkD#, sEb
| kkD#, sEb
|sD#, (kkD#), sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
Line 102: Line 102:
| comma-narrow aug unison, little minor 2nd  
| comma-narrow aug unison, little minor 2nd  
| kD#, lEb
| kD#, lEb
|kD#, kEb
| kD#, kEb
|-
|-
| 6
| 6
Line 113: Line 113:
| minor 2nd
| minor 2nd
| Eb
| Eb
|Eb
| Eb
|-
|-
| 7
| 7
Line 124: Line 124:
| classic minor 2nd
| classic minor 2nd
| KEb
| KEb
|KEb
| KEb
|-
|-
| 8
| 8
Line 135: Line 135:
| on minor 2nd
| on minor 2nd
| OEb
| OEb
|SEb
| SEb
|-
|-
| 9
| 9
Line 146: Line 146:
| neutral 2nd
| neutral 2nd
| UEb/uE
| UEb/uE
|UEb/uE
| UEb/uE
|-
|-
| 10
| 10
Line 157: Line 157:
| off major 2nd
| off major 2nd
| oE
| oE
|sE
| sE
|-
|-
| 11
| 11
Line 168: Line 168:
| classic/comma-narrow major 2nd
| classic/comma-narrow major 2nd
| kE
| kE
|kE
| kE
|-
|-
| 12
| 12
Line 179: Line 179:
| major 2nd
| major 2nd
| E
| E
|E
| E
|-
|-
| 13
| 13
Line 190: Line 190:
| large major 2nd
| large major 2nd
| LE
| LE
|KE
| KE
|-
|-
| 14
| 14
Line 201: Line 201:
| supermajor 2nd
| supermajor 2nd
| SE
| SE
|SE
| SE
|-
|-
| 15
| 15
| 250.000
| 250.000
| 81/70,  15/13
| 81/70,  15/13
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| ^<sup>3</sup>M2, <br />v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| trupmajor 2nd,<br />trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| ^<sup>3</sup>E, <br />v<sup>3</sup>F
| HM2, hm3
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| hypermajor 2nd, hypominor 3rd
| HE, hF
| HE, hF
|UE, uF
| UE, uF
|-
|-
| 16
| 16
Line 223: Line 223:
| subminor 3rd
| subminor 3rd
| sF
| sF
|sF
| sF
|-
|-
| 17
| 17
Line 234: Line 234:
| little minor 3rd
| little minor 3rd
| lF
| lF
|kF
| kF
|-
|-
| 18
| 18
Line 245: Line 245:
| minor 3rd
| minor 3rd
| F
| F
|F
| F
|-
|-
| 19
| 19
Line 256: Line 256:
| classic minor 3rd
| classic minor 3rd
| KF
| KF
|KF
| KF
|-
|-
| 20
| 20
Line 267: Line 267:
| on minor third
| on minor third
| OF
| OF
|SF
| SF
|-
|-
| 21
| 21
Line 278: Line 278:
| neutral 3rd
| neutral 3rd
| UF/uF#
| UF/uF#
|UF/uF#
| UF/uF#
|-
|-
| 22
| 22
Line 289: Line 289:
| off major 3rd
| off major 3rd
| oF#
| oF#
|sF#
| sF#
|-
|-
| 23
| 23
Line 300: Line 300:
| classic major 3rd
| classic major 3rd
| kF#
| kF#
|kF#
| kF#
|-
|-
| 24
| 24
Line 311: Line 311:
| major 3rd
| major 3rd
| F#
| F#
|F#
| F#
|-
|-
| 25
| 25
Line 322: Line 322:
| large major 3rd
| large major 3rd
| LF#
| LF#
|KF#
| KF#
|-
|-
| 26
| 26
Line 333: Line 333:
| supermajor 3rd
| supermajor 3rd
| SF#
| SF#
|SF#
| SF#
|-
|-
| 27
| 27
Line 344: Line 344:
| hypermajor 3rd, hypo 4th
| hypermajor 3rd, hypo 4th
| HF#, hG
| HF#, hG
|UF#, uG
| UF#, uG
|-
|-
| 28
| 28
Line 355: Line 355:
| sub 4th
| sub 4th
| sG
| sG
|sG
| sG
|-
|-
| 29
| 29
Line 366: Line 366:
| little 4th
| little 4th
| lG
| lG
|kG
| kG
|-
|-
| 30
| 30
Line 377: Line 377:
| perfect 4th
| perfect 4th
| G
| G
|G
| G
|-
|-
| 31
| 31
Line 388: Line 388:
| comma-wide 4th
| comma-wide 4th
| KG
| KG
|KG
| KG
|-
|-
| 32
| 32
Line 399: Line 399:
| on 4th
| on 4th
| OG
| OG
|SG
| SG
|-
|-
| 33
| 33
Line 410: Line 410:
| uber 4th / neutral 4th
| uber 4th / neutral 4th
| UG
| UG
|UG
| UG
|-
|-
| 34
| 34
Line 421: Line 421:
| classic aug 4th, sub dim 5th
| classic aug 4th, sub dim 5th
| kkG#, sAb
| kkG#, sAb
|SG#, (kkG#), sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
Line 433: Line 433:
| comma-narrow aug 4th, little dim 5th
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, lAb
|kG#, kAb
| kG#, kAb
|-
|-
| 36
| 36
Line 444: Line 444:
| aug 4th, dim 5th
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|G#, Ab
| G#, Ab
|-
|-
| 37
| 37
Line 455: Line 455:
| large aug 4th, comma-wide dim 5th
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| LG#, KAb
|KG#, KAb
| KG#, KAb
|-
|-
| 38
| 38
Line 467: Line 467:
| super aug 4th, classic dim 5th
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, KKAb
|SG#, SAb, (KKAb)
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
Line 478: Line 478:
| unter 5th / neutral 5th
| unter 5th / neutral 5th
| uA
| uA
|uA
| uA
|-
|-
| 40
| 40
Line 489: Line 489:
| off 5th
| off 5th
| oA
| oA
|sA
| sA
|-
|-
| 41
| 41
Line 500: Line 500:
| comma-narrow 5th
| comma-narrow 5th
| kA
| kA
|kA
| kA
|-
|-
| 42
| 42
Line 511: Line 511:
| perfect 5th
| perfect 5th
| A
| A
|A
| A
|-
|-
| 43
| 43
Line 522: Line 522:
| large fifth
| large fifth
| LA
| LA
|KA
| KA
|-
|-
| 44
| 44
Line 533: Line 533:
| super fifth
| super fifth
| SA
| SA
|SA
| SA
|-
|-
| 45
| 45
Line 544: Line 544:
| hyper fifth, hypominor 6th
| hyper fifth, hypominor 6th
| HA, hBb
| HA, hBb
|UA, uBb
| UA, uBb
|-
|-
| 46
| 46
Line 555: Line 555:
| superminor 6th
| superminor 6th
| sBb
| sBb
|sBb
| sBb
|-
|-
| 47
| 47
Line 566: Line 566:
| little minor 6th
| little minor 6th
| lBb
| lBb
|kBb
| kBb
|-
|-
| 48
| 48
Line 577: Line 577:
| minor 6th
| minor 6th
| Bb
| Bb
|Bb
| Bb
|-
|-
| 49
| 49
Line 588: Line 588:
| classic minor 6th
| classic minor 6th
| kBb
| kBb
|kBb
| kBb
|-
|-
| 50
| 50
Line 599: Line 599:
| on minor 6th
| on minor 6th
| oBb
| oBb
|sBb
| sBb
|-
|-
| 51
| 51
Line 610: Line 610:
| neutral 6th
| neutral 6th
| UBb, uB
| UBb, uB
|UBb, uB
| UBb, uB
|-
|-
| 52
| 52
Line 621: Line 621:
| off major 6th
| off major 6th
| oB
| oB
|sB
| sB
|-
|-
| 53
| 53
Line 632: Line 632:
| classic major 6th
| classic major 6th
| kB
| kB
|kB
| kB
|-
|-
| 54
| 54
Line 640: Line 640:
| major 6th
| major 6th
| B
| B
|M6
| M6
|major 6th
| major 6th
|B
| B
|B
| B
|-
|-
| 55
| 55
Line 654: Line 654:
| large major 6th
| large major 6th
| LB
| LB
|KB
| KB
|-
|-
| 56
| 56
Line 665: Line 665:
| supermajor 6th
| supermajor 6th
| SB
| SB
|SB
| SB
|-
|-
| 57
| 57
| 950.000
| 950.000
| 121/70
| 121/70
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| ^<sup>3</sup>M6, <br />v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| trupmajor 6th,<br />trudminor 7th
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| ^<sup>3</sup>B, <br />v<sup>3</sup>C
| HM6, hm7
| HM6, hm7
| hypermajor 6th, hypominor 7th
| hypermajor 6th, hypominor 7th
| HB, hC
| HB, hC
|UB, uC
| UB, uC
|-
|-
| 58
| 58
Line 687: Line 687:
| subminor 7th
| subminor 7th
| sC
| sC
|sC
| sC
|-
|-
| 59
| 59
Line 698: Line 698:
| little minor 7th
| little minor 7th
| lC
| lC
|kC
| kC
|-
|-
| 60
| 60
Line 709: Line 709:
| minor 7th
| minor 7th
| C
| C
|C
| C
|-
|-
| 61
| 61
Line 720: Line 720:
| classic/comma-wide minor 7th
| classic/comma-wide minor 7th
| KC
| KC
|KC
| KC
|-
|-
| 62
| 62
Line 731: Line 731:
| on minor 7th
| on minor 7th
| OC
| OC
|SC
| SC
|-
|-
| 63
| 63
Line 742: Line 742:
| neutral 7th, hypo dim 8ve
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, hDb
|UC/uC#, uDb
| UC/uC#, uDb
|-
|-
| 64
| 64
Line 753: Line 753:
| off major 7th, sub dim 8ve
| off major 7th, sub dim 8ve
| oC#, sDb
| oC#, sDb
|sC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
Line 764: Line 764:
| classic major 7th, little dim 8ve
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, lDb
|kC#, kDb
| kC#, kDb
|-
|-
| 66
| 66
Line 775: Line 775:
| major 7th, dim 8ve
| major 7th, dim 8ve
| C#, Db
| C#, Db
|C#, Db
| C#, Db
|-
|-
| 67
| 67
Line 786: Line 786:
| large major 7th, comma-wide dim 8ve
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| LC#, KDb
|KC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
Line 797: Line 797:
| supermajor 7th, classic dim 8ve
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, KKDb
|SC#, SDb, (KKDb)
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
Line 808: Line 808:
| hypermajor 7th, unter 8ve, hypo 8ve
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| HC#, uD, hD
|UC#, uDb, uD
| UC#, uDb, uD
|-
|-
| 70
| 70
Line 819: Line 819:
| sub 8ve, off 8ve
| sub 8ve, off 8ve
| sD, oD
| sD, oD
|sD
| sD
|-
|-
| 71
| 71
Line 830: Line 830:
| comma-narrow 8ve, little 8ve
| comma-narrow 8ve, little 8ve
| kD, lD
| kD, lD
|kD
| kD
|-
|-
| 72
| 72
Line 841: Line 841:
| perfect octave
| perfect octave
| D
| D
|D
| D
|}
|}
<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament. For lower limits see [[Table of 72edo intervals]].


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 977: Line 975:


Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* -1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* &minus;1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* -2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* &minus;2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* +3 degrees  (the trup ring)corrects 4/3 to 11/8 via 33/32
* +3 degrees  (the trup ring)corrects 4/3 to 11/8 via 33/32
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
Line 1,016: Line 1,014:
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning Error
|-
|-
Line 1,057: Line 1,055:
| 3.59
| 3.59
|}
|}
* 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  
72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  


=== Commas ===
=== Commas ===
Line 1,063: Line 1,061:


{| class="commatable wikitable center-1 center-2 right-4"
{| class="commatable wikitable center-1 center-2 right-4"
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br />Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
Line 1,279: Line 1,277:
| Jacobin comma
| Jacobin comma
|}
|}
<references/>


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,288: Line 1,285:
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
|-
|-
! Periods<br>per 8ve
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />Ratio*
! Temperaments
! Temperaments
|-
|-
Line 1,367: Line 1,364:
|-
|-
| 2
| 2
| 21\72<br>(19\72)
| 21\72<br />(19\72)
| 316.7<br>(283.3)
| 316.7<br />(283.3)
| 6/5<br>(13/11)
| 6/5<br />(13/11)
| [[Bikleismic]]
| [[Bikleismic]]
|-
|-
| 2
| 2
| 23\72<br>(13\72)
| 23\72<br />(13\72)
| 383.3<br>(216.7)
| 383.3<br />(216.7)
| 5/4<br>(17/15)
| 5/4<br />(17/15)
| [[Wizard]] / lizard / gizzard
| [[Wizard]] / lizard / gizzard
|-
|-
Line 1,385: Line 1,382:
|-
|-
| 3
| 3
| 19\72<br>(5\72)
| 19\72<br />(5\72)
| 316.7<br>(83.3)
| 316.7<br />(83.3)
| 6/5<br>(21/20)
| 6/5<br />(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
|-
| 4
| 4
| 19\72<br>(1\72)
| 19\72<br />(1\72)
| 316.7<br>(16.7)
| 316.7<br />(16.7)
| 6/5<br>(105/104)
| 6/5<br />(105/104)
| [[Quadritikleismic]]
| [[Quadritikleismic]]
|-
|-
| 8
| 8
| 34\72<br>(2\72)
| 34\72<br />(2\72)
| 566.7<br>(33.3)
| 566.7<br />(33.3)
| 168/121<br>(55/54)
| 168/121<br />(55/54)
| [[Octowerck]] / octowerckis
| [[Octowerck]] / octowerckis
|-
|-
| 8
| 8
| 35\72<br>(1\72)
| 35\72<br />(1\72)
| 583.3<br>(16.7)
| 583.3<br />(16.7)
| 7/5<br>(100/99)
| 7/5<br />(100/99)
| [[Octoid]] / octopus
| [[Octoid]] / octopus
|-
|-
| 9
| 9
| 19\72<br>(3\72)
| 19\72<br />(3\72)
| 316.7<br>(50.0)
| 316.7<br />(50.0)
| 6/5<br>(36/35)
| 6/5<br />(36/35)
| [[Ennealimmal]] / ennealimnic
| [[Ennealimmal]] / ennealimnic
|-
|-
| 9
| 9
| 23\72<br>(1\72)
| 23\72<br />(1\72)
| 383.3<br>(16.7)
| 383.3<br />(16.7)
| 5/4<br>(105/104)
| 5/4<br />(105/104)
| [[Enneaportent]]
| [[Enneaportent]]
|-
|-
| 12
| 12
| 23\72<br>(1\72)
| 23\72<br />(1\72)
| 383.3<br>(16.7)
| 383.3<br />(16.7)
| 5/4<br>(100/99)
| 5/4<br />(100/99)
| [[Compton]] / comptone
| [[Compton]] / comptone
|-
|-
| 18
| 18
| 19\72<br>(1\72)
| 19\72<br />(1\72)
| 316.7<br>(16.7)
| 316.7<br />(16.7)
| 6/5<br>(105/104)
| 6/5<br />(105/104)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 24
| 24
| 23\72<br>(1\72)
| 23\72<br />(1\72)
| 383.3<br>(16.7)
| 383.3<br />(16.7)
| 5/4<br>(105/104)
| 5/4<br />(105/104)
| [[Hours]]
| [[Hours]]
|-
|-
| 36
| 36
| 23\72<br>(1\72)
| 23\72<br />(1\72)
| 383.3<br>(16.7)
| 383.3<br />(16.7)
| 5/4<br>(81/80)
| 5/4<br />(81/80)
| [[Decades]]
| [[Decades]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct.


== Scales ==
== Scales ==
Line 1,670: Line 1,667:
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
== Notes ==
<references group="note" />


[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/72edo"